Hypergeometric BKP conjecture for the generalized BGW tau-function

Let $\operatorname{DP}$ be the set of strict partitions. A hypergeometric BKP tau-function has the form

$$\tau=\sum_{\lambda\in\operatorname{DP}}r_\lambda Q_\lambda({\bf t}/2)Q_\lambda({\bf t}^*/2),$$

where $r_\lambda=\prod_{j=1}^{\ell(\lambda)}\prod_{m=1}^{\lambda_j}r(m)$ for a function $r(z)$. Let $\tau_{BGW}({\bf t}/2,N)$ be the generalized BGW partition function, with parameter $N$. Generalized BGW hypergeometric BKP conjecture. The partition function $\tau_{BGW}({\bf t}/2,N)$ is a hypergeometric tau-function of the BKP hierarchy, with

$$r(z)=\hbar\frac{(2z-1)^2-4N^2}{16},$$

and $t_k^*=2\delta_{k,1}$. This conjecture is known in the special BGW case $N=0$ from the preceding BGW expansion conjecture, while the generalized statement remains open in the source.